We ended up doing a coaching session with Gravity Nelson - it really helped! I had been using my brakes all wrong, trying to "pump" them in order to avoid skidding out, but it seems gradual and steady pressure works better. It will be a long time before I'm really comfortable with the actual mountain bike tracks, I didn't even get to learning how to go around berms - and those seem to be all over the place here.
At least we finally got a bike rack installed so we're not limited to what's right on our doorstep :-) We're in Hanmer Springs for a few days, and plan to do a family ride. It seems like bike trails are often promoted as one of the main "things to do" when visiting, and it sure beats renting bikes everywhere we go.